\(V^{\prime }_{\alpha ,0}\simeq V^{\prime }_{\alpha ,1}\)  for all \(\alpha \in \mathbb {C}\) .
For convenience we denote \(V^{\prime }_{\alpha , \beta }:= V_{\alpha , \beta }\)  if \(V_{\alpha , \beta }\)  is irreducible. We have the following:
Theorem 3.2 [1]} 
Any nontrivial irreducible uniformly bounded \(\mathrm {Vir}\) -module is isomorphic to \(V^{\prime }_{\alpha , \beta }\)  for some \(\alpha , \beta \in .\) 
Now, consider the Lie algebra \(\mathrm {Vir} \otimes B\) , where \(B\)  is finitely generated commutative associative unital algebra over \(. For \) B = t, t-1]\( the following result was proved in \cite {krg}:\)
Theorem 3.3 Let \(V\)  be an irreducible uniformly bounded \(\mathrm {Vir} \otimes t, t^{-1}]\) -module. Then \(V\)  is a single point evaluation module.

Savage generalised the above result for any finitely generated commutative associative unital algebra over \( \cite {sav}:\begin{thm}Let V be an irreducible uniformly bounded \mathrm {Vir} \otimes B-module. Then V is a single point evaluation module.\end{thm}\) We now move our attention to \(n \ge 2\)  case.
Let us recall the tensor field module on a torus \(T(U, {\mathbf {\gamma }}): = t^{{\mathbf {\gamma }}} A_n \otimes U\) , where \(U\)  is a finite dimensional irreducible \(\mathfrak {gl}_n\)  representation and
\({\mathbf {\gamma }} \in n\) , which has the following \(W_n\) -module structure:
\(t^{\bf m}d_i. (t^{{\mathbf {\gamma }}+ {\bf {s}}}) = (\gamma _i + s_i) t^{{\mathbf {\gamma }} + {\bf s}+ {\bf m}} \otimes u + \sum _{j = 1}^{n}{m_j t^{{\mathbf {\gamma }} + {\bf s}+ {\bf m}}} \otimes E_{j i} u,\) 
where \({\bf m, s} \in \mathbb {Z}^n,\)  \(u \in U,\)  \(1 \le i \le n\)  and \(E_{ji}\)  is the \(n \times n\)  matrix whose \((j,i)\) th entry is 1 and all others are zero. Let \(V\)  be standard representation of \(\mathfrak {gl}_n\)  of dimension \(n\) . We know that for \( k = 0, \ldots , n\) , \(\Lambda ^{k} \,V\)  is an irreducible finite dimensional representation of \(\mathfrak {gl}_n\) . Then the tensor field modules \( T(\Lambda ^{k} \,V, {\mathbf {\gamma }}) =: \Omega ^{k}({\mathbf {\gamma }})\)  are also called modules of differential \(k\) -forms and they form the de Rham complex:
\(\Omega ^{0} ({\mathbf {\gamma }}) \xrightarrow{} \Omega ^{1} ({\mathbf {\gamma }}) \xrightarrow{} \cdots \xrightarrow{} \Omega ^{n}({\mathbf {\gamma }}) .\) 
The differential map \(d: \Omega ^{k}({\mathbf {\gamma }}) \rightarrow \Omega ^{k+1}({\mathbf {\gamma }})\)  is \(W_n\) -module map for \(k = 0, \ldots , n-1\) , so we have that \(\Omega ^{k}({\mathbf {\gamma }})\)  is reducible \(W_n\) -module for \(k = 1, \dots , n-1\)  and \(\Omega ^{0}({\mathbf {\gamma }}), \Omega ^{n}({\mathbf {\gamma }})\) 
are reducible iff \({\mathbf {\gamma }} \in \mathbb {Z}^n\) . We have the following:
Theorem 3.4  [2]}
Let \(V\)  be a non-trivial irreducible uniformly bounded \(W_n\) -module. Then one of the following statements hold:

Either \(V \cong T(U, {\mathbf {\beta }})\) , where \(U\)  is an irreducible finite dimensional \(\mathfrak {gl}_n\) -module which is not an exterior power of a standard \(n\) -dimensional \(\mathfrak {gl}_n\) -module and \({\mathbf {\gamma }} \in n\) , or

\(V \cong d(\Omega ^k)({\mathbf {\beta }}) \subseteq \Omega ^{k+1}({\mathbf {\beta }})\) , where \(0 \le k <n\)  and \({\mathbf {\beta }} \in n\) .


Now, consider an \(\mathfrak {L}\) -module \(V\) . We say the action of \(A_n\)  is associative on \(V\)  if \(t^{{\bf m}} t^{{\bf s}}.v = t^{{\bf m+s}}.v \)  for all \({\bf m, s} \in \mathbb {Z}^n\)  and \(v \in V\) . The following result was proved in [3]}:
Theorem 3.5 Let \(V\)  be an irreducible \(\mathfrak {L}\) -module with finite dimensional weigh spaces with an associative action of \(A_n\)  and \(t^{{\bf 0}}.v = v\)  for all \(v \in V\) . Then \(V \cong T(U, {\mathbf {\gamma }})\) , where \(U\)  is an irreducible
finite dimensional representation of \(\mathfrak {gl}_n\)  and \({\mathbf {\gamma }} \in n\) , with the following actions:
\(t^{\bf m}d_i. (t^{{\mathbf {\gamma }}+ {\bf {s}}}) = (\gamma _i + s_i) t^{{\mathbf {\gamma }} + {\bf s}+ {\bf m}} \otimes u + \sum _{j = 1}^{n}{m_j t^{{\mathbf {\gamma }} + {\bf s}+ {\bf m}}} \otimes E_{j i} u,\) 
\(t^{{\bf m}}. t^{{\mathbf {\gamma }} + {\bf s}} \otimes u = t^{{\mathbf {\gamma }} + {\bf m + s}} \otimes u.\) 

In the following result [4]} it is proved that if \(V\)  is an irreducible uniformly bounded \(\mathfrak {L}\) -module and if \(t^{\bf 0}\)  acts by a non-zero scalar then the action of \(A_n\)  is associative on \(V\) . In the case where \(t^{\bf 0}\)  acts trivially on \(V\) ,
\(A_n\)  acts trivially on \(V\) , and so \(V\)  is an irreducible uniformly bounded \(W_n\) -module. More precisely:
Theorem 3.6 [4]} 
Let \(V\)  be an irreducible uniformly bounded \(\mathfrak {L}\) -module. Then

If \(t^{\bf 0}\)  acts as a non-zero scalar then \(A_n\)  acts associatively on \(V\)  and \( V \cong T(U, {\mathbf {\gamma }})\) , where \(U\)  is an irreducible finite dimensional \(\mathfrak {gl}_n\) -module and \({\mathbf {\gamma }} \in n\) .

If \(t^{\bf 0}\)  acts as a zero then \(A_n. V = 0\)  and hence \(V\)  is an irreducible \(W_n\) -module.


The following result deals with the classification of irreducible Harish-Chandra \(\mathfrak {L}(B)\) -modules under some conditions:
Theorem 3.7 [6]}
Let \(V\)  be an irreducible weight module for \(\mathfrak {L}(B)\)  with finite dimensional weight spaces. Assume that the action of \(A_n\otimes B\)  is associative on \(V\)  and \(t^{\bf 0} .v=v\)  for all \(v \in V.\)  Then \(V\)  is a single point evaluation \(\mathfrak {L}(B)\)  module and
hence an irreducible \(\mathfrak {L}\) -module.

Finally, we recall some results on \(\mathrm {HVir}\)  which will play an important role in this paper.
For the Lie algebra HVir, the subalgebra spanned by \(\lbrace  x_0, I(0), C_{D}, C_{DI}, C_{I}\rbrace \)  plays a role of Cartan subalgebra. For \(a, b, c \in  consider HVir-module \) W(a,b, c)\( with a basis \) {wi | i Z}\( and the following action:\)\(x_j. w_i = (a+i+bj)w_{i+j}, I(j).w_{i}= c w_{i+j}, C_D .w_i = C_{DI}.w_i = C_I . w_i = 0.\)\( It is known that \) W(a,b,c)\( is reducible iff \) a Z\(, \) b\( is either 0 or 1 and \) c =0\(. Let \) W'(a,b,c)\( be a unique non-trivial irreducible sub-quotient of\) W(a,b,c)\(. The following is an important result from \cite [Theorem 4.4]{lrkz}.\begin{thm} Let V be an irreducible HVir-module with uniformly bounded weight spaces. Then V \cong W^{\prime }(a,b,c) for some a,b,c \in .\end{thm}Now consider \)HVir\( be the Lie algebra \) HVirC̏D C̏DI C̏I\( with the Cartan subalgebra \) x̏0 Ȉ(0)\(. It is easy to see that as a Lie algebra\)HVir\( is isomorphic to \) (j Z D(, js) k Ztks )\(, where \) 0 s Zn\( and \)  n\( such that \) (  |  s) 0\( with\) xi D(, is)(  |  s)\( and \) I(i) tis(  |  s).\(Taking into account of the above isomorphism we denote the Lie algebra \) (j Z D(, js) k Ztks )\( by \)HVir(, s).\(This paper is concerned with the classification of all the irreducible Harish-Chandra modules of \) LS, n(B)\(.The following lemma will be used in this section. The proof of this lemma follows along similar lines as that of proof of Lemma 3.2 in cite{kgx}.\begin{lemma}Let V be an irreducible \mathfrak {L}(B)-module such that u.v = 0 for some u \in U(A_n \otimes B) and some non-zero vector v in V. Then u is locally nilpotent on V.\end{lemma}\subsection {Reduction to \mathfrak {L}(B) or W_n (B)-modules}We start with the following:\begin{prop} Let V be an irreducible uniformly bounded \mathfrak {L}_{S, n}(B)-module.\begin{enumerate}[label= (0)]\item Let (S \otimes t̏^{\bf 0}) \otimes 1 act non-tivially on V. Then there exists a subspace \overset{\circ }{S} of S of co-dimension 1 such that (\overset{\circ }{S} \otimes A_n) \otimes B acts trivially on V.In particular V is an irreducible \mathfrak {L}(B)-module.\item If (S \otimes t̏^{\bf 0}) \otimes 1 acts trivially, then V is an irreducible W_n (B)-module.\end{enumerate}\end{prop}{\begin{xmlelement*}{proof}Note that (S \otimes t̏^{\bf 0} )\otimes 1 is central in \mathfrak {L}_{S,n}(B), hence its elements act as scalars on V. Since (S \otimes t̏^{\bf 0}) \otimes 1 acts non-trivially on V, there exists a co-dimension one subspace \overset{\circ }{S} of S such that (\overset{\circ }{S} \otimes t̏^{\bf 0}) \otimes 1.V=0.Let us choose a {\mathbf {\gamma }} \in \mathbb {C}^n such that ({\mathbf {\gamma }}, {\bf s})\ne 0 for all non-zero {\bf s} \in \mathbb {\mathbb {Z}}^n. Fix a non-zero {\bf m} \in \mathbb {Z}^n and some y \in \overset{\circ }{S}. Now consider the Heisenberg-Virasoro algebra \mathrm {HVir}({\mathbf {\gamma }}, {\bf m},y)= \mathrm {span} \lbrace D({\mathbf {\gamma }} \,| i {\bf m}), y\otimes t^{i {\bf m}} \otimes 1 \, \, | \,\, i \in \mathbb {Z} \rbrace . For {\bf r} \in \mathbb {Z}^n consider the uniformly bounded \mathrm {HVir}({\mathbf {\gamma }}, {\bf m}, y)-module \displaystyle {\bigoplus _{i \in \mathbb {Z}}V_{\lambda + {\bf r} +i {\bf m}}}. Now, following the proof of Theorem 3.3(2) of \cite {kgx}, which relies on Theorem \ref {thm1}, we find a least positive integer N such that (y\otimes t^{\bf m} \otimes 1)^N.V=0. But then we haveD({\mathbf {\gamma }},{\bf k-m})\otimes b.(y\otimes t^{\bf m} \otimes 1)^N.V= ({\mathbf {\gamma }} | {\bf m})Ny\otimes t^{\bf k}\otimes b.(y\otimes t^{\bf m}\otimes 1)^{N-1}.V=0, for all {\bf k} \in \mathbb {Z}^n, b \in B. In particular, there exists a non-zero u \in V such that y \otimes t^{\bf r} \otimes b.u=0 for all {\bf r} \in \mathbb {Z}^n , b \in B. Then consider the set W: =\lbrace v \in V: y \otimes t^{\bf r} \otimes b.v=0 for all {\bf r} \in \mathbb {Z}^n, b \in B\rbrace , which is a non-zero submodule of V so must be V itself.This completes the proof of (1). Similar argument as that of proof of (1) will work for (2).\end{xmlelement*}}\subsection {t^{{\bf 0}} \otimes 1 acting as an identity}By above proposition we can assume \) V\( to be an irreducible uniformly bounded \) L(B)\(-module. In the next proposition we work with the assumption that \) t0 1\( acting non-zero scalar say c. By taking a Lie algebra automorphism of \) An\( which assigns \) ts tsc\(, we may assume \) c = 1\(. We have the following:\)
Theorem 3.8 
Let \(V\)  be an irreducible uniformly bounded module for \(\mathfrak {L}(B)\) . If \(t^{\bf 0}\otimes 1\)  acts as an identity on \(V\) , then \(V\)  is a single point evaluation module.

We will use Theorem 4.5 of [6]}, by which we will be done if we show that \(A_n \otimes B\)  acts associatively on \(V\) , i.e., \(t^{\bf r}\otimes b_1 t^{{\bf s}} \otimes b_2.v=t^{{\bf r+s}}\otimes b_1b_2.v\)  for all \({\bf r,s} \in \mathbb {Z}^n, b_1,b_2 \in B, v \in V.\) 
First as \(1\otimes B\)  is central in \(\mathfrak {L}(B)\)  there exists a linear map \(\psi :B \rightarrow \mathbb {C}\)  such that \(\psi (1)=1\) . If \(b \in \mathrm {Ker} \,\psi \) , then \(1 \otimes b. v = 0\)  for all \(v \in V\) .
Claim : \((t^{\bf r} \otimes bb^{\prime }).V=0\) , for all \(b \in \mathrm {Ker}\, \psi , b^{\prime } \in B,{\bf r} \in \mathbb {Z}^n\) . In particular, \(\mathrm {Ker} \,\psi \)  is a maximal ideal of \(B\) .
Let \({\mathbf {\gamma }} \in \mathbb {C}^n\)  be such that \(({\mathbf {\gamma }} \,|\, {\bf s})\ne 0\)  for all non-zero \({\bf s} \in \mathbb {Z}^n\) . Fix \(0 \ne {\bf m} \in \mathbb {Z}^n\)  and \(b \in \mathrm {Ker}\, \psi \)  and consider the Lie algebra \(\mathrm {HVir}({\mathbf {\gamma }},{\bf m},b) = \mathrm {span} \, \lbrace D({\mathbf {\gamma }}, i{\bf m}), t^{i {{\bf m}}}\otimes b \, \, | \,\, i \in \mathbb {Z} \rbrace \) .
Now using similar arguments as that of Theorem 3.2 (2) of [4]}, find a smallest positive integer \(N_0\)  such that \((t^{\bf m} \otimes b)^{N_0}.V=0\)  for all \({ \bf m} \in \mathbb {Z}^n.\)  Then we have
\(D({\mathbf {\gamma }}, {\bf k -m}) \otimes b^{\prime }.(t^{\bf m} \otimes b)^{N_0}.V = ({\mathbf {\gamma }} \,|\, {\bf s})N_0 \, (t^{\bf k} \otimes bb^{\prime }) \, (t^{\bf m} \otimes b)^{N_0 -1}.V = 0 ,\) 
for all \({\bf k} \in \mathbb {Z}^n\)  and \(b^{\prime } \in B.\) 
Hence we have \(V^{\prime } = \lbrace v \in V \mid t^{\bf r} \otimes bb^{\prime }.v= 0 \, \forall \, \, {\bf r} \in \mathbb {Z}^n, \, \forall \, b^{\prime } \in B \rbrace \)  is a non-zero submodule of \(V\) . This proves the claim.
Note that \(b -\psi (b)1 \in \mathrm {Ker}\, \psi \) , therefore from the above claim it follows that, \(t^{\bf r} \otimes b.v=\psi (b)t^{\bf r}.v\)  for all \({\bf r} \in \mathbb {Z}^n, b \in B, v \in V\) .
Since \(\mathrm {Ker} \,\,\psi \)  is maximal ideal in \(B\)  of co-dimension one there exists a non-zero algebra homomorphism \(\eta :B \rightarrow \mathbb {C}\)  such that \(\eta =c \psi \) . Now \(\eta (1)=\eta (1)\eta (1)\)  with the facts \(\psi (1)=1\)  and \(\eta \)  non-zero gives us \(c=1.\)  Hence we have,
\(t^{\bf r} \otimes b.v=\eta (b)t^{\bf r}.v,\) 

for all \({\bf r} \in \mathbb {Z}^n, b \in B, v \in V\)  and for some algebra homomorphism \(\eta \) . Hence to prove the associativity of \(A_n \otimes B\)  we have to prove the associativity of \(A_n \otimes 1.\)  Now using the techniques of
proof of Theorem 3.2 (1) of [4]}, which involve Lemma , we can find a non-zero \(w \in V\)  such that \(t^{\bf m}.t^{\bf n}.w=t^{{\bf m+n}}.w\)  for all \({\bf m,n} \in \mathbb {Z}^n\) . Finally note that \(W=\lbrace v \in V: t^{\bf m}.t^{{\bf n}}.v=t^{{\bf m+n}}.v\rbrace  \)  is a non-zero \(\mathfrak {L}(B)\) -submodule of \(V\) . Hence the proof is complete.

\(t^{{\bf 0}} \otimes 1\)  acting trivially
Now we concentrate on the case where \(t^{{\bf 0}}\otimes 1\)  acts trivially on \(V\) . By Proposition (2) \(V\)  is an irreducible \(W_n (B)\) -module. We have the following:
Theorem 3.9 
Let \(V\)  be a uniformly bounded irreducible module for \(W_n (B)\) 
with finite dimensional weight spaces. Then there exists a co-finite ideal \(J\)  of \(B\)  such that \(W_n \otimes J.V=0.\)

As \(V\)  is irreducible, we have
\(V=\displaystyle {\bigoplus _{{\bf m} \in \mathbb {Z}^n}}V_{{\mathbf {\alpha }}+ {\bf m}},\)  where \({\mathbf {\alpha }} \in \mathbb {C}^n\)  with dim \(V_{\mathbf {\alpha }} \ne 0\)  and \(V_{{\mathbf {\alpha }}+ {\bf m}} = \lbrace  v \in V: D({\bf u}, {\bf 0})v=({\bf u}, {\mathbf {\alpha }}+ {\bf m})v \)  for all \({\bf u} \in \mathbb {C}^n \rbrace \) .
Let \(N \in \mathbb {N}\)  such that \(\mathrm {dim}\,(V_{\alpha +m}) \le N\)  for all \( {\bf m} \in \mathbb {Z}^n\) .
For \({\bf 0} \ne {\bf r} \in \mathbb {Z}^n\) , let \(I_{(1,{\bf r})} := \lbrace  b \in B : D({\bf e_1, r}) \otimes b.V_\alpha = 0\rbrace \) . Then \(I_{(1,{\bf r})}\)  is the kernel of the linear map
\( B \rightarrow \mathrm {Hom} (V_{\mathbf {\alpha }}, V_{{\mathbf {\alpha }}+ {\bf r}}),b \rightarrow (v \rightarrow D({\bf e_1, r})bv),\) 
and hence we have \(\mathrm {dim}(B/I_{(1,{\bf r})}) \le N^2\) . Note that \([D({\bf e_j, 0}) \otimes s, D({\bf e_1, r}) \otimes b] = r_jD({\bf e_1, r}) \otimes bs\)  implies that \(I_{(1,{\bf r})}\)  is an ideal of B for all \(r \ne 0\) .
Claim 1 : \(I_{(1,{\bf e_1})}I^j_{(1,{\bf e_2})} \subseteq I_{(1, {\bf e_1} + j {\bf e_2})}\) , for all \(j \ge 1.\)
Note that \(D({\bf e_1, e_1 + e_2}) \otimes bc = -[D({\bf e_1, e_1}) \otimes b, D({\bf e_1, e_2}) \otimes c].\)  This implies that the claim is
true for \(j = 1.\)  Now it will follow by induction that the claim 1 is true for all \(j \ge 1.\)
Claim 2 : \( I_{(1,{\bf e_1})}{I^{N^2}_{(1,{\bf e_2}) }}\subseteq I_{(1, {\bf e_1} + j {\bf e_2})}\)  for all \(j \ge 1\) .
Fix some \(j \in \mathbb {N}\)  and consider the chain of subspaces
\(\frac{B}{I_{(1, {\bf e_1} + j {\bf e_2})}} \supseteq \frac{I_{(1,{\bf e_1})}+I_{(1, {\bf e_1} + j {\bf e_2})}}{I_{(1, {\bf e_1} + j {\bf e_2})}} \supseteq \frac{I_{(1,{\bf e_1})}I_{(1,{\bf e_2})}+I_{(1, {\bf e_1} + j {\bf e_2})}}{I_{(1, {\bf e_1} + j {\bf e_2})}} \supseteq \frac{I_{(1,{\bf e_1})}I^2_{(1,{\bf e_2})}+I_{(1, {\bf e_1} + j {\bf e_2})}}{I_{(1, {\bf e_1} + j {\bf e_2})}} \supseteq \cdots \) 
Since \(\mathrm {dim}( B/I_{(1, {\bf e_1} + j {\bf e_2})}) \le N^2,\)  there exists an \(l \le N^2\)  such that \(I_{(1,{\bf e_1})}I^l_{(1,{\bf e_2})} + I_{(1, {\bf e_1} + j {\bf e_2})} =I_{(1,{\bf e_1})}I^{l+1}_{(1,{\bf e_2})} + I_{(1, {\bf e_1} + j {\bf e_2})}.\)  This implies that, \(I_{(1,{\bf e_1})}I^m_{(1,{\bf e_2})} + I_{(1, {\bf e_1} + j {\bf e_2})} =I_{(1,{\bf e_1})}I^l_{(1,{\bf e_2})} + I_{(1, {\bf e_1} + j {\bf e_2})}\)  for all \(m \ge l\)  and \(l \le N^2.\) 
Now \(I_{(1,{\bf e_1})}I^j_{(1,{\bf e_2})} \subseteq I_{(1, {\bf e_1} + j {\bf e_2})} \)  implies that \(I_{(1,{\bf e_1})}\, I^l_{(1,{\bf e_2})} + I_{(1, {\bf e_1} + j {\bf e_2})} = I_{(1, {\bf e_1} + j {\bf e_2})}\)  for \(1 \le j \le l\) . Hence we have \(I_{(1,{\bf e_1})}I^ l_{(1,{\bf e_2})}+ I_{(1, {\bf e_1} + j {\bf e_2})} = I_{(1, {\bf e_1} + j {\bf e_2})}, \, \forall \, j \ge 1,\)  this proves the Claim 2.
Claim 3 : \(I_{(1,{\bf e_1})}I^{N^2}_{(1,{\bf e_2})}I^{N^2}_{(1,-{\bf e_2})} \subseteq I_{(1,{\bf e_1}+r {\bf e_2})}\)  for all \(r \in \mathbb {Z} .\) 
Note that \(I_{(1,{\bf e_1})}I^{N^2}_{(1,{\bf e_2})}I_{(1,-{\bf e_2})} \subseteq I_{(1,{\bf e_1+je_2-e_2})} \)  for all \(j \ge 1,\)  because
\([D({\bf e_1 , e_1} + j\,{\bf e_2})\otimes ab, D({\bf e_1}, -{\bf e_2})\otimes c] = -D({\bf e_1, e_1} + j{ \bf e_2 - e_2})\otimes abc \) 
for all \(a \in I_{(1,{\bf e_1})}, b \in I^{N^2}_{(1,{\bf e_2})}, c \in I_{(1,-{\bf e_2})}.\)  Hence by induction we have \(I_{(1,{\bf e_1})}I^{N^2}_{(1,{\bf e_2})}I^k_{(1,-{\bf e_2}) }\subseteq I_{(1,{\bf e_1}+j\,{\bf e_2} -k \,{\bf e_2})}\)  for all \(j \ge 1, k \ge 1.\)  Now with the same procedure as like claim 2 we
have \(I_{(1,{\bf e_1})}I^{N^2}_{(1,{\bf e_2})}I^{N^2}_{(1,-{\bf e_2})} \subseteq I_{(1, {\bf e_1}+j{\bf e_2}-k{\bf e_2})}\)  for all \(j,k \ge 1.\)  Hence claim 3 follows.
Now it is easy to see with similar arguments that \(I_{(1,{\bf e_1})}\displaystyle {\prod _{j=2}^{n}I^{N^2}_{(1,{\bf e_j})}I^{N^2}_{(1,-{\bf e_j})}} \subseteq I_{(1, {\bf e_1+r})}\)  for all \({\bf r} \in \mathbb {Z}^{n-1}.\)
Claim 4: \(I_{(1,{\bf e_1})}I^{N^2}_{(1,2{\bf e_1})} \displaystyle {\prod _{j=2}^{n}I^{N^2}_{(1,{\bf e_j})}I^{N^2}_{(1,-{\bf e_j})}}\subseteq I_{(1,{\bf e_1}+2k{\bf e_1}+r {\bf e_2})}\)  for all \({\bf r} \in \mathbb {Z}^{n-1}\) , \(k \ge 1.\) 
Note that \(I_{(1,{\bf e_1})}I_{(1,2{\bf e_1})}\displaystyle {\prod _{j=2}^{n}I^{N^2}_{(1,{\bf e_j})}I^{N^2}_{(1,-{\bf e_j})}} \subseteq I_{(1, {\bf e_1}+2{\bf e_1}+r {\bf e_2})}\)  for all \({\bf r} \in \mathbb {Z}^{n-1}\) , because
\( [D({\bf e_1, e_1 + r}) \otimes ab, D({\bf e_1}, 2{\bf e_1})\otimes c] = D({\bf e_1, e_1} + 2{\bf e_1 + r}) \otimes abc.\) 
Hence by induction we have \(I_{(1,{\bf e_1})}I^k_{(1,2{\bf e_1})}\displaystyle {\prod _{j=2}^{n}I^{N^2}_{(1,{\bf e_j})}I^{N^2}_{(1,-{\bf e_j})}} \subseteq I_{(1,{\bf e_1}+2k{\bf e_1+r})}\)  for all \(r \in \mathbb {Z}, k \ge 1.\)  Now applying the method like claim 2 we have claim 4.
Claim 5:
Set \( I_1 = I_{(1,{\bf e_1})}I^{N^2}_{(1,2{\bf e_1})}I^{N^2}_{(1,-3e_1)}\displaystyle {\prod _{j=2}^{n}I^{N^2}_{(1,{\bf e_j})}I^{N^2}_{(1,-{\bf e_j})}} \subseteq I_{(1,{\bf e_1}+2k{\bf e_1}-3l{ \bf e_1+r})}\) 
for all \({\bf r} \in \mathbb {Z}^{n-1}, k,l \ge 1.\)  This claim is clear with the previous method.
Note that the set \(\lbrace 1 + 2k - 3l : k, l \in \mathbb {N} \rbrace  = \mathbb {Z}.\)  Hence \(I_1 \subseteq I_{(1,{\bf r})},\)  for all \({\bf r} \in \mathbb {Z}^n \setminus \lbrace 0\rbrace \) .
Claim 6 : \(D({\bf e_1, r}) \otimes J_1.V_\alpha = 0,\)  for all \({\bf r} \in \mathbb {Z}^n\) , \(J_1=I_1^2\) .
Note that \(J_1 \subseteq I_1,\)  so \(D({\bf e_1, r}) \otimes J_1.V_\alpha = 0,\)  for all \({\bf r} \in \mathbb {Z}^n \setminus \lbrace 0\rbrace .\)  For \({\bf r = 0},\)  just note
that
\([D({\bf e_1, e_1}) \otimes a, D({\bf e_1, -e_1})\otimes b] = 2D({\bf e_1, 0}) \otimes ab.\) 
This proves the claim.
In a similar way we get co-finite ideals \(J_i\)  such that \(D({\bf e_i, r}) \otimes J_i.V_\alpha = 0\)  for all \({\bf r} \in \mathbb {Z}^n, 1 \le i \le n.\)  Then for \(J := J_1J_2 \cdots J_n\)  we have \(W_n \otimes J.V_\alpha = 0\) 
and \(W = \lbrace v \in V : W_n \otimes J.v = 0\rbrace \)  is a non-zero submodule of \(V\) , so must be equal to \(V\) .
To proceed further, we need to recall the definitions of \(A_nW_n\) -module structure and an \(A_n\) -cover for any \(W_n\) -module \(M\)  (see [2]} for more details).
Definition 3.10 A \(W_n\) -module \(M\)  is said to have an \(A_nW_n\) -module structure if it is also a module for an associative algebra \(A_n\)  and satisfies the following:
\(X(fv) = (Xf)v + f(Xv), X \in W_n, f \in A_n, v \in M.\) 

In other words, \(M\)  is called an \(A_n W_n\) -module if \(M\)  is an \(\mathfrak {L}\) -module with an associative action of \(A_n\) , i.e., \(t^{{\bf 0}}.v = v\)  and \(t^{{\bf m+n}}.v = t^{{\bf m}}t^{{\bf n}}.v\)  for all \(v \in M\) .
It is easy to see that \(W_n\)  has an \(A_n W_n\) -module structure with an \(A_n\)  action given as follows:
\(t^{{\bf m}}. t^{{\bf s}} d_i = t^{{\bf m+s}} d_i.\) 
Definition 3.11 An \(A_n\) -cover of a \(W_n\) -module \(M\)  is the subspace \(\widehat{M} \subset \mathrm {Hom}(A_n, M)\)  spanned by the set \(\lbrace \psi (X,v):X\in W_n, v \in M\rbrace \) , where \(\psi (X,v) \in \mathrm {Hom}(A_n,M)\)  is given by
\(\psi (X,v)(f)=(fX)v \)  for all \(f \in A_n\) .

An \(A_n\) -cover of a \(W_n\) -module \(M\)  acquires an \(A_n W_n\) -module structure:
\(X.\psi (Y, v) &= \psi ([X,Y], v) + \psi (Y, X.v)\\g.\psi (Y, v) &= \psi (g Y, v), \,\, X,Y \in W_n, g \in A_n. v \in M,\) 

and it is useful in the following sense [2]} Theorem 4.8:
Theorem 3.12 Let \(V\)  be an irreducible uniformly bounded module for \(W_n\) . Then its \(A_n\)  cover \(\widehat{V}\)  is a uniformly bounded \(\mathfrak {L}\) -module.

Theorem 3.13 
Any uniformly bounded irreducible \(W_n\otimes B\) -module is a single point evaluation module.

Let \(V\)  be an irreducible uniformly bounded \(W_n (B)\) -module. Then by Theorem REF , \(V\)  becomes a uniformly bounded
irreducible module for \(W_n (R)\) , for some finite dimensional commutative associative unital algebra \(R\) . Now,
\(V\)  is a uniformly bounded \(W_n\) -module. Hence its \(A_n\)  cover \(\widehat{V} := \mathrm {span}\lbrace \phi (x,v):x \in W_n, v \in V ,\,\,\, \phi (x,v) \in \mathrm {Hom}(A_n,V)\rbrace \)  is a uniformly bounded \(\mathfrak {L}\) -module.
Then, consider the vector space \(\widehat{V} \otimes R\) , to which an \(\mathfrak {L}(R)\) -module structure can be given as follows:
\(D({\bf u,r})\otimes b.\phi (x,m) \otimes b^{\prime } &= \phi ([D({\bf u,r}),x],m) \otimes bb^{\prime }+\phi (x,D({\bf u,r})\otimes b.m) \otimes b^{\prime },\\t^{\bf r}\otimes b.\phi (x,m) \otimes b^{\prime } &= \phi (t^{\bf r} x,m) \otimes bb^{\prime },\) 

for all \({\bf u}\in \mathbb {C}^n,{\bf r} \in \mathbb {Z}^n, x \in W_n,m \in V, b,b^{\prime } \in R.\)  It is easy to see that this action
defines a module structure on \(\widehat{V} \otimes R\) . We also note the action of \(A_n \otimes R\)  is associative on \(\widehat{V} \otimes R\) .
Now observe that \(\eta :\widehat{V} \otimes R \rightarrow V \)  defined by \(\eta (\phi (x, m)b) = (x \otimes b).m\)  is a
surjective \(W_n (R)\) -module map. As \(\widehat{V}\)  is uniformly bounded and R is finite dimensional, the
module \(\widehat{V} \otimes R\)  is a uniformly bounded \(\mathfrak {L}(R)\) -module. Therefore there is a composition
series for the \(\mathfrak {L}\otimes R\) -module \(\widehat{V} \otimes R.\)  Consider the composition series
\( 0 = V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdots \subseteq V_s = \widehat{V} \otimes R, \) 
where \(V_i/V_{i-1}\)  is an irreducible uniformly bounded \(\mathfrak {L}\otimes R\) -module for \(1 \le i \le s\) . Let \(1 \le k \le s\) 
be the smallest positive integer such that \(\eta (V_k) \ne 0\) . Then \(\eta : V_k/V_{k-1} \rightarrow V\)  is
a surjective \(W_n (R)\) -module map. Therefore \(V\)  is isomorphic to an irreducible
quotient of \(V_k/V_{k-1}\) , hence a single point evaluation module by Theorem 4.5 of [6]}.
Note that all the proofs of our results in this section go through if we assume \(n= 1\) .

Final result: Uniformly bounded case
We assimilate the information from Proposition , Theorem REF , and Theorem REF  in the following
Theorem 3.14 
Let \(n \ge 1\) , then any non-trivial irreducible uniformly bounded \(\mathfrak {L}_{S,n}(B)\) -module can be considered as an irreducible \(\mathfrak {L}(B)\)  or \(W_n (B)\) -module. Furthermore, as a \(\mathfrak {L}(B)\)  or \(W_n (B)\) -module V is a single point evaluation module.

Remark 3.15 For n = 1, Theorem REF  was proved in [13]} for more general Lie algebras \(\mathfrak {L}_{S, 1}\) , where \(S\)  is a Lie superalgebra with an even diagonalisable derivation.
In particular, Theorem REF  generalises the result of [13]} when \(S\)  is assumed to be a finite dimensional abelian Lie algebra.

In the next section, we will deal with the irreducible \(\mathfrak {L}_{S,n}(B)\) -modules with finite but non-uniformly bounded weight spaces. We will see that the classification of those modules hinges on the classification result
of irreducible uniformly bounded \(\mathfrak {L}_{S,n}\) -modules. In particular, the following result will be used repeatedly in the next section:
Theorem 3.16 Let \(W\)  be a non-zero \(\mathfrak {L}_{S,n}\) -module (\(n \ge 1\) ) with uniformly bounded weight spaces. Then, \(W\)  contains a non-zero irreducible \(\mathfrak {L}_{S,n}\) -module. Furthermore,

If \( {\bf 0} \notin \mathrm {Supp}(W)\) , then \(W\)  is an infinite dimensional \(\mathfrak {L}_{S,n}\) -module and contains a non-trivial irreducible \(\mathfrak {L}_{S,n}\) -submodule.

If \( {\bf 0} \in \mathrm {Supp}(W)\) , then either \(W\)  is finite dimensional trivial \(\mathfrak {L}_{S,n}\) -module or \(W\)  is an infinite dimensional \(\mathfrak {L}_{S,n}\) -module and \(W/W_1\)  contains a
non-trivial irreducible \(\mathfrak {L}_{S,n}\) -module, where \(W_1\)  is the maximal finite dimensional trivial \(\mathfrak {L}_{S,n}\) -submodule of \(W\) .


The existence of a non-zero irreducible \(\mathfrak {L}_{S,n}\) -module is guaranteed by the fact that \(W\)  has a composition series of finite length. The results (1) and (2) follow by applications of the Theorem REF , Theorem REF , Theorem , Theorem REF , and Theorem REF  respectively.

Non-uniformly bounded \(\mathfrak {L}_{S, n}(B)\) -modules
In this section we classify all the irreducible Harish-Chandra \(\mathfrak {L}_{S, n}(B)\) -modules with non-uniformly bounded weight spaces. We need to introduce the following
Highest weight modules
We start with recalling some result proved for \(\mathrm {Vir}\)  in [15]}. Let \(\mathrm {Vir}^- \oplus \mathrm {Vir}^0 \oplus \mathrm {Vir}^+\)  be standard triangular decomposition of \(\mathrm {Vir}\) , where
\(\mathrm {Vir}^- = \mathrm {span}\lbrace x_i \mid i \in \mathbb {Z}_{<0}\rbrace \) , \(\mathrm {Vir}^0 = \mathrm {span} \lbrace C, x_0\rbrace \)  and \(\mathrm {Vir}^+ = \mathrm {span}\lbrace x_i \mid i \in \mathbb {N}\rbrace \) .
Let
\(\mathrm {Vir}(B)^{-} \oplus \mathrm {Vir}(B)^{0} \oplus \mathrm {Vir}(B)^{+}\)  be a triangular decomposition of \(\mathrm {Vir}(B)\) , where
\(\mathrm {Vir}(B)^{-} = \mathrm {Vir}^{-} \otimes B\) , \(\mathrm {Vir}(B)^{0} = \mathrm {Vir}^{0} \otimes B\)  and \(\mathrm {Vir}(B)^{+} = \mathrm {Vir}^{+} \otimes B.\)
Definition 4.1 A \(\mathrm {Vir}(B)\) -module \(V\)  is called a highest weight module if it is a weight module and there exists a non-zero weight vector \(v\)  such that \(\mathrm {Vir}(B)^+ . v = 0\)  and \(U(\mathrm {Vir})(B) = V\) .

Let \({\psi }\)  be the one dimensional representation of \(\mathrm {Vir}(B)^{0}\) , where \(\psi \in \mathrm {Hom}(\mathrm {Vir}(B)^{0}, \) . Make \({\psi }\)  as a \(\mathrm {Vir}(B)^{0} \oplus \mathrm {Vir}(B)^{+}\)  with a trivial action of
\(\mathrm {Vir}(B)^{+} \) on it. Then consider Verma module
\(M(\psi ) := \mathrm {Ind}_{\mathrm {Vir}(B)^{0} \oplus \mathrm {Vir}(B)^{+}}^{U(\mathrm {Vir}(B))} {\psi }.\) 
Let \(V(\psi )\)  be the unique irreducible quotient. It is proved in [15]} (Proposition 5.1) that \(V(\psi )\)  is Harish-Chandra module iff there exist a cofinite ideal \(I\)  of \(B\)  such that \(\mathrm {Vir} \otimes I. V(\psi ) = 0\) .
The following result also proved in [15]} (Theorem 5.3)
Theorem 4.2 Every irreducible Harish-Chandra \(\mathrm {Vir}(B)\) -module with non-uniformly bounded weight spaces is a highest weight module and is a tensor product of finitely many irreducible single point generalised evaluation highest weight modules.

One can define highest weight modules for \(\mathfrak {L}_{S,1}\)  as above. We have similar result holds for \(\mathfrak {L}_{S,1}(B)\) :
Theorem 4.3 Every irreducible Harish-Chandra \(\mathfrak {L}_{S,1}(B)\) -module with non-uniformly bounded weight spaces is a highest weight module and is a tensor product of finitely many irreducible single point generalised evaluation highest weight modules.

Follows from Lemma 2.7 of [13]} and by similar techniques that of proof of Theorem 5.3 in [15]}.
We will now assume \(n \ge 2\) .
Let \(\mathfrak {L}_{S,n} = \bigoplus _{{\bf m} \in \mathbb {Z}^n}{{(\mathfrak {L}_{S,n})}_{{\bf m}}}\)  be the root space decomposition of \(\mathfrak {L}_{S, n}\)  with respect to \(H\) , where
\({(\mathfrak {L}_{S,n})}_{\bf m} ={\left\lbrace \begin{array}{ll}H \ltimes (S \otimes t^{{\bf 0}}) & \text{if ${\bf m = 0}$}\\\big (\bigoplus _{i =1}^n{t̏^{\bf m} d_i} \big ) \ltimes (S \otimes t^{\bf m}) & \text{otherwise}.\end{array}\right.}\) 
Let \(M\)  be a subgroup of \(\mathbb {Z}^n\)  and \({\mathbf {\beta }} \in \mathbb {Z}^n\)  such that \(\mathbb {Z}^n = M \oplus \mathbb {Z}{\mathbf {\beta }}\) . Consider a triangular decomposition of \({ \mathfrak {L}^-_{S,n}} \oplus {(\mathfrak {L}_{S,n})}_{M} \oplus {\mathfrak {L}^+_{S,n}}\)  of \(\mathfrak {L}_{S,n}\) , where
\({\mathfrak {L}^-_{S,n}} = \bigoplus _{d \in \mathbb {N}}{(\mathfrak {L}_{S,n})}_{M - d{\mathbf {\beta }}}, {\mathfrak {L}^+_{S,n}} = \bigoplus _{d \in \mathbb {N}} {(\mathfrak {L}_{S,n})}_{M + d{\mathbf {\beta }}}, {(\mathfrak {L}_{S,n})}_{M} = \bigoplus _{{\bf r} \in M}{{(\mathfrak {L}_{S,n})}_{\bf r}}.\) 
Definition 4.4 An \(\mathfrak {L}_{S,n}\) -module \(V\)  is called a highest weight module if it is a weight module and there exists a non-zero weight vector \(v\)  with the property that \({\mathfrak {L}_{S,n}}^+ . v = 0\)  and \(U(\mathfrak {L}_{S,n}).v = V\) .

Let \(Y\)  be an irreducible module for \({(\mathfrak {L}_{S,n})}_{M}\) . Make \(Y\)  as \({(\mathfrak {L}_{S,n})}_{M} \oplus {\mathfrak {L}^+_{S,n}}\) -module with a trivial action of \({\mathfrak {L}_{S,n}}^{+}\)  on it. The generalised Verma module for \(\mathfrak {L}_{S,n}\)  is defined as
\(V_{\mathfrak {L}_{S,n}}(Y, {{\mathbf {\beta }}}, M) := \mathrm {Ind}_{({\mathfrak {L}_{S,n})}_{M} \oplus {\mathfrak {L}^+_{S,n}}}^{U(\mathfrak {L}_{S,n})} Y.\) 
Now by the standard arguments \(V_{\mathfrak {L}_{S,n}}(Y, {\mathbf {\beta }}, M)\)  contains a unique irreducible sub-quotient \(L_{\mathfrak {L}_{S,n}}(Y, {\mathbf {\beta }}, M)\) . By the results of Zhao and Billig [20]}, it follows that \(L_{\mathfrak {L}_{S, n}}(Y, {\mathbf {\beta }}, M)\)  has finite dimensional weight spaces if \(Y\)  is a uniformly bounded
exp-polynomial module for \({({\mathfrak {L}}_{S,n})}_M\)  (see [20]} for definition). We have the following result:
Theorem 4.5 
Let \(V\)  be an irreducible \(\mathfrak {L}_{S,n}\) -module with finite but not uniformly bounded weight spaces. Then \(V \cong L_{\mathfrak {L}_{S,n}}(Y, {\mathbf {\beta }}, M)\)  for some \(Y, {\mathbf {\beta }}, M\)  defined as above.

Proof follows by exactly similar argument as that of proof of Theorem 4.3 in [4]}.
Similarly, let \(\mathfrak {L}_{S,n}(B) = \bigoplus _{{\bf m} \in \mathbb {Z}^n}{{\mathfrak {L}_{S,n}(B)}_{\bf m}}\)  be the root space decomposition of \(\mathfrak {L}_{S,n}(B)\)  with respect to \(H\) , where
\({\mathfrak {L}_{S, n}(B)}_{\bf m} ={\left\lbrace \begin{array}{ll}(H \ltimes (S \otimes t^{{\bf 0}}))\otimes B & \text{if ${\bf m = 0}$}\\\big ( \big ( \bigoplus _{i =1}^n{ t̏^{\bf m} d_i} \big ) \ltimes S \otimes t^{\bf m} \big ) \otimes B & \text{otherwise}.\end{array}\right.}\) 
As above we define the generalised Verma module
\(V_{\mathfrak {L}_{S,n}(B)}(Y, {\mathbf {\beta }}, M) : = \mathrm {Ind}_{{\mathfrak {L}_{S,n}(B)}_M \oplus {\mathfrak {L}_{S,n}(B)}^{+}}^{U({\mathfrak {L}_{S,n}(B)})} Y\) 
and \(L_{\mathfrak {L}_{S,n}(B)}(Y, {\mathbf {\beta }}, M)\)  be its unique irreducible quotient. Again by [20]}, it follows that
\(L_{\mathfrak {L}_{S,n}(B)}(Y, {\mathbf {\beta }}, M)\)  has finite dimensional weight spaces if \(Y\)  is a uniformly bounded \({\mathfrak {L}_{S,n}(B)}_M\) -module.
Automorphisms of \(\mathfrak {L}_{S,n}\) :
Let \(GL_n (\mathbb {Z})\)  be the group of invertible matrix with entries in \(\mathbb {Z}\) . Let \(A = (a_{ij}) \in GL_n (\mathbb {Z})\)  with \(\mathrm {det}(A) = \pm 1\) . Then \(A\)  defines an Lie algebra automorphism
\(\Phi _{A} : \mathfrak {L}_{S,n} \rightarrow \mathfrak {L}_{S,n}\)  as
\(\Phi _{A} (t^{{\bf m}} d_j) = \sum _{p =1}^{n}{b_{jp} t^{{\bf m}A^{T}} d_p}, \,\,\, \Phi _{A}(s \otimes t^{{\bf m}}) = s \otimes t^{{\bf m} A^{T}}, \) 
where \(1 \le j \le n, {\bf m} \in \mathbb {Z}^n, s \in S\)  and \(A^{-1} = B = (b_{i j})\) . The term up to a change of variable will always mean twisting of an \(\mathfrak {L}_{S,n}\)  action on its module by a suitable automorphism \(\Phi _{A}\) .
We need to recall the definition of a generalised highest weight vector. Let \({\mathbf {\beta }}, {\mathbf {\gamma }} \in \mathbb {Z}^n\) . We denote \({\mathbf {\beta }} \ge {\mathbf {\gamma }}\)  if \(\beta _i \ge \gamma _i\)  for all \(1 \le i \le n\)  and \({\mathbf {\beta }} > {\mathbf {\gamma }}\)  if \(\beta _i > \gamma _i\)  for all \(1 \le i \le n\) .
Definition 4.6 Let \(M\)  be any \(\mathfrak {L}_{S,n}\) -module. A vector \(v \in M\)  is called a generalised highest weight vector if there exists \(N \in \mathbb {N}\)  such that \((\mathfrak {L}_{S,n})_{\mathbf {\alpha }}.v = 0\)  for all \({\mathbf {\alpha }} > (N, \ldots , N)\) .


A key result
Let \({\mathbf {\lambda }} \in n\)  and let \(W\)  be any weight module for \(\mathfrak {L}_{S,n}(B)\)  or \(\mathfrak {L}_{S,n}\) . Let \(K\)  be any subgroup of \(\mathbb {Z}^n\) , then we define \(W[{\mathbf {\lambda }} + K] := \bigoplus _{{\bf k} \in K}{W_{{\mathbf {\lambda }}+{\bf k}}}\) . In particular we have
\(\mathfrak {L}_{S,n}(B) [K] = \bigoplus _{{\bf k} \in K}{(\mathfrak {L}_{S,n} (B))_{{\bf k}}}.\) 
The following theorem plays a key role in the classification:
Theorem 4.7 
Let \(V\)  be an irreducible Harish-Chandra \(\mathfrak {L}_{S,n}(B)\) -module with non-uniformly bounded weight spaces. Then \(V\)  is an \(\mathfrak {L}_{S,n} (B)\) -highest weight module.

It is enough to prove that there exists a non-zero weight vector \(z\)  with weight \(\mathbf {\lambda }\)  and an integer \(p\)  with the property that
\(\mathfrak {L}_{S, n}(B)[{(p + j}) \mathbf {\beta }+ K] .z = 0\)  for all \(j \ge 1\) , where \(\mathbb {Z}^n = \mathbb {Z}\mathbf {\beta }\oplus K\) . Then by irreducibilty of \(V\)  and PBW theorem, it will imply that the \(\mathrm {Supp}(V) \subseteq \lbrace \mathbf {\lambda }+ (p - j)\mathbf {\beta }+ K: j \in \mathbb {Z}_{+} \rbrace \) . Hence we will have
\(V = U(\mathfrak {L}_{S, n} (B)).Z\) , where \(Z = U(\mathfrak {L}_{S, n} (B) [K]).z = V[{\mathbf {\lambda }+ p \mathbf {\beta }+ K}]\)  is the highest weight space of \(V\) .
Now, since \(V\)  is irreducible, there exists \({\mathbf {\mu }} \in n\)  such that Supp \(V \subseteq {\mathbf {\mu }} + \mathbb {Z}^n\) . Choose a \({\mathbf {\gamma }}\in n\)  with the property that \((\mathbf {{\mathbf {\gamma }}}|\bf {m}) \ne 0\)  for all \(\bf {m}\in \mathbb {Z}^n \setminus \lbrace 0\rbrace \) . Then consider \(V\)  as \(\mathrm {Vir}(\mathbf {\mathbf {\gamma }})\) -module and by exactly similar argument as that of Theorem 4.3 of [4]} we get \(\bf {m}\in \mathbb {Z}^n\)  such that \(m_1=0\)  and
\(\mathrm {dim} V_{{\mathbf {\mu }}-\bf {m}}>(n+ \mathrm {dim}\, S)(\mathrm {dim}V_{{{\mathbf {\mu }}+\bf {e}}_1}+\sum _{i=2}^{n}\mathrm {dim}V_{{\mathbf {\mu }}+{\bf {e}}_1+{\bf {e}}_i})\) 
Let \({\bf {q}}_1={\bf {m}}+{\bf {e}}_1\) , \({\bf {q}}_i={\bf {m}}+{\bf {e}}_1+{\bf {e}}_i\)  for \( i=2,\dots ,n\) . For simplification, let \(\lambda := {{\mathbf {\mu }}}-\bf {m}\) . Then by (4.3) we have non-zero vector \(v \in V_\lambda \)  such that \(D({\bf {e}}_i, {\bf {q}}_j). v= 0\)  and \(S\otimes t^{{\bf {q}}_j}.v=0\)  for all \(i,j=1,\dots ,n\) . It follows that \(v\)  is an \(\mathfrak {L}_{S,n}\) -generalised highest weight vector. Now consider the \(\mathfrak {L}_{S,n}\) -submodule \(W:=U(\mathfrak {L}_{S,n}).v\)  of \(V\) . Let \(\overset{\circ }{W}\)  be the submodule of \(W\)  containing trivial vectors We will prove at the end of this section (Proposition
REF ) that \(\overset{\circ }{W} = \lbrace {\bf 0}\rbrace .\), i.e., \(\overset{\circ }{W} = \lbrace v \in W \mid \mathfrak {L}_{S,n} . v = 0\rbrace \) .
Consider the quotient module \(\overline{W}:= W/ \overset{\circ }{W}\) .
Claim 1: An \(\mathfrak {L}_{S,n}\) -module \(\overline{W}\)  contains no non-zero trivial vectors.
Suppose contrary, there exists a trivial vector \(\bar{w} = w + \overset{\circ }{W}\) , with \(w \notin \overset{\circ }{W}\)  such that \(\mathfrak {L}_{S,n}w \in \overset{\circ }{W}\) . But as \(\bar{w}\)  is a trivial vector, it must lie in the zero weight space of \(\overline{W}\) . It implies that the weight of \(w\)  is also a zero
weight vector considered as a weight vector of \(W\) .
But we have \((\mathfrak {L}_{S,n})_{\pm {\bf e_i}}.w \in \overset{\circ }{W}\) , and as all vectors \(\overset{\circ }{W}\)  have weight zero (considered as a weight vectors of W), it follows that \((\mathfrak {L}_{S,n})_{\pm {\bf e_i}}. w = 0 \) . Hence we have \(\mathfrak {L}_{S,n}.w = 0\) , which is absurd. This proves claim 1. Let \(\bar{v}\)  denote the image of \(v\)  under the natural map from \(W\)  to \(\overline{W}\) . Then we have \(\overline{W} =U( \mathfrak {L}_{S,n}). (\bar{v})\) .
Claim 2: \(\overline{W}\)  contains a highest weight \(\mathfrak {L}_{S,n}\) -module.
By claim 1, \(\overline{W}\)  contains no non-zero trivial vectors. We have the following:

Every vector in \(\overline{W}\)  is a generalised highest weight vector.

For any \(\mathbf {\alpha }\in \mathbb {N}^n\)  and any \(0\ne \bar{w} \in \overline{W}\) , \( (\mathfrak {L}_{S,n})_{-\mathbf {\alpha }}.\bar{w} \ne 0\) .

For \(\eta \in \)  Supp\((\overline{W})\) , and any \(\mathbf {\alpha }\in \mathbb {N}^n\) , the set \(\lbrace \ell \in \mathbb {Z}| \eta +\ell \mathbf {\alpha }\in \)  Supp\((\overline{W})\rbrace =(-\infty , s]\)  for some \(s\in \mathbb {Z}_{+}\) .

Upto a change of variables we have \(\lambda +\mathbf {\alpha }\notin \)  Supp\((\overline{W})\)  for any non-zero \(\mathbf {\alpha }\in \mathbb {Z}_{+}^n\) .

Proof of above statements are exactly similar to Lemma 3.3, 3.4, 3.5, 3.6 of [25]} respectively. By (d) it follows that \(\overline{W}\)  is not a uniformly bounded \(\mathfrak {L}_{S,n}\) -module. So let \(\mathfrak {L}_{S,n}(K,\mathbf {\beta },X)\)  be its irreducible sub-quotient. Now as every element of \(\overline{W}\)  is a generalised highest weight vector, replacing \(\bar{v}\)  by a suitable vector if necessary we can assume that
\(\mathbf {\lambda }-\mathbb {Z}_{+}\mathbf {\beta }+K \setminus \lbrace {\bf 0}\rbrace  \subseteq \)  Supp\((\overline{W})\) .
We also have the following properties (see Lemma 3.7, [25]})

\(\mathbf {\lambda }+\mathbf {\alpha }+K \nsubseteq \)  Supp(\(\overline{W}\) ) for all \({\mathbf {\alpha }} \in \mathbb {Z}_{+}^n \setminus \lbrace {\bf 0}\rbrace \) .

\(\exists ~ {\mathbf {\nu }}\in \mathbb {N}^n\)  such that \(K:= \lbrace {\bf {y}}\in \mathbb {Z}^n : ({\mathbf {\nu }|\bf {y})}= 0\rbrace \) .

\(\mathbf {\lambda }+\bf {y}\in \)  Supp(\(\overline{W}\) ) \(\forall ~ {\bf {y}}\in \mathbb {Z}^n\)  with \((\mathbf {\nu }|\bf {y})<0\) .

Subclaim:
\(\lbrace \mathbf {\lambda }+k\mathbf {\beta }+ K: k\in \mathbb {N}\rbrace \bigcap \)  Supp\((\overline{W})\ne \phi \)  for only finitely many \(k\in \mathbb {N}\) .
Assume contrary, so we get a strictly increasing sequence of \(k_i\in \mathbb {N}\)  such that
\(\lbrace \mathbf {\lambda }+k_i\mathbf {\beta }+K: i\in \mathbb {N}\rbrace \bigcap \)  Supp\((\overline{W})\ne \phi \) .
Now by choosing sufficiently large \(k_i\in \mathbb {N}\)  and taking a subsequence of \(\lbrace k_i\in \mathbb {N}\rbrace \)  if necessary we can write
\( |\lbrace \mathbf {\lambda }+k_r\mathbf {\beta }+K\rbrace \bigcap \lbrace \mathbf {\lambda }+\mathbb {Z}_{+}^n\rbrace |>1\) 
\(|\lbrace \mathbf {\lambda }+k_{r-1}\mathbf {\beta }+K\rbrace \bigcap \lbrace \mathbf {\lambda }+\mathbb {Z}_{+}^n\rbrace |>0\) 
By equation REF , we have that \(W[\mathbf {\lambda }+k_{r}\mathbf {\beta }+K]\)  is a non-uniformly bounded \(\mathfrak {L}_{S,n}[K]\) -module. Now following the argument of the proof of Lemma 3.7 line by line we arrive at a contradiction. This completes the proof of subclaim.
Let \(j_0\in \mathbb {Z}_{+}\)  be the largest integer such that
\( \lbrace \mathbf {\lambda }+j_0\mathbf {\beta }+K \rbrace \bigcap \mathrm {Supp}(\overline{W})\ne \phi .\) 
Consider \(\overline{W}[\mathbf {\lambda }+j_0\mathbf {\beta }+K]\)  as \(\mathfrak {L}_{S,n}[K]\) -module. Observe that \(\overline{W}[\mathbf {\lambda }+j_0\mathbf {\beta }+K]\)  is a uniformly bounded \(\mathfrak {L}_{S,n}[K]\) -module. Otherwise it would contain an irreducible \(\mathfrak {L}_{S,n}[K]\bigcap \mathrm {Vir}(\mathbf {\mathbf {\gamma }})\)  sub-quotient say \(\bar{Y}\)  which is not uniformly bounded. But then the unique irreducible quotient of
\(\mathrm {Ind}_{U((\mathfrak {L}_{S,n}[K] \cap \mathrm {Vir}({\mathbf {\gamma }}))+({\mathfrak {L}_{S,n}}^+ \bigcap \mathrm {Vir}(\mathbf {\mathbf {\gamma }})))}^{U(\mathrm {Vir}(\mathbf {\mathbf {\gamma }}))}\bar{Y}\) 
is not a Harish-Chandra module which will imply that \(\overline{W}\)  is not a Harish-Chandra module.
Now as \(\overline{W}[\mathbf {\lambda }+j_0\mathbf {\beta }+K]\)  is uniformly bounded \(\mathfrak {L}_{S,n}[K]\) -module, let \(\overline{Z}\)  be its non-zero irreducible submodule. It is easy to see that the module \(\bar{M} := U(\mathfrak {L}_{S,n}).\overline{Z}\)  is the required highest weight \(\mathfrak {L}_{S,n}\) -module
contained in \(\overline{W}\) . This completes the proof of claim 2. Now we need to deal with two cases:
Case 1: \({\bf 0} \notin \mathrm {Supp}({\overline{W}})\) . Note that in this case all the zero weight vectors of \(W\)  lie in \(\overset{\circ }{W}\) . Let \(\overline{Z}\)  be its non-trivial irreducible submodule of \(\overline{W}[\mathbf {\lambda }+j_0\mathbf {\beta }+K]\) .
Let \(0 \ne \bar{z} \in \overline{Z}\)  be any weight vector. Let \(z\)  be any preimage of \(\bar{z}\)  under the surjective \(\mathfrak {L}_{S,n}[K]\) -map \(\Phi : W[{\mathbf {\lambda }}+j_0{\mathbf {\beta }}+K] \rightarrow \overline{W}[{\mathbf {\lambda }}+j_0{\mathbf {\beta }}+K]\) . Let us define \(M := U(\mathfrak {L}_{S,n}).z\) . Let \(M^{\prime }\)  be the maximal
\(\mathfrak {L}_{S,n}\) -submodule of \(M\)  with property that \(M_{{\mathbf {\gamma }}}^{\prime } = 0\)  for all \(\mathbf {\gamma }\in \mathrm {Supp}(\overline{Z})\) . The existence of \(M^{\prime }\)  follows from the fact that \(M\)  is a cyclic module and equation REF . Observe also that \(M \cap \overset{\circ }{W} \subseteq M^{\prime }\) .
Let us consider the restriction of map \(\Phi \)  on an \(\mathfrak {L}_{S,n}[K]\) -module \(M \lbrace K\rbrace  := U(\mathfrak {L}_{S,n} [K]).z\) , which we denote by \(\psi \) . Then \(\psi : M \lbrace K\rbrace  \mapsto \overline{Z}\)  is a surjective \(\mathfrak {L}_{S,n} [K]\) -module map defined by \(\psi (u.z) = u.\bar{z}\) , where \(u \in \mathfrak {L}_{S,n} [K]\) . We have
\(\mathrm {Kernel}(\psi ) = M\lbrace K\rbrace  \cap \overset{\circ }{W}\) .
Claim 3: \(M/M^{\prime }\)  is an irreducible \(\mathfrak {L}_{S,n}\) -module.
Assume on the contrary that there exists a proper non-zero \(\mathfrak {L}_{S, n}\) -submodule \(M_1\)  of \(M\)  containing \(M^{\prime }\)  such that \((M_1)_{{\mathbf {\gamma }}} \ne 0\)  where \({\mathbf {\gamma }} \in \mathrm {Supp} (\overline{Z})\) .
Let \(0 \ne z_1 \in (M_1)_{{\mathbf {\gamma }}}\) . Then there exists \(u \in U(\mathfrak {L}_{S,n})\)  such that \(u.z = z_1\) .
Subclaim: \(u \in U(\mathfrak {L}_{S,n}[K])\) .
Let us assume that the subclaim is not true. Then \(u\)  must be of the form \(u_1 u_2 u_3\) , where \(u_1 \in U_0(\mathfrak {L}_{S,n}^-)\) , \(u_2 \in U(\mathfrak {L}_{S,n}[K])\)  and \(u_3 \in U_0(\mathfrak {L}_{S,n}^+)\) . Here, for any Lie algebra \(\mathfrak {g}\) , \(U_0(\mathfrak {g})\)  denotes the argumentation ideal of
\(U(\mathfrak {g})\) . But by equation REF , it follows that \(u_3.z = 0\)  unless weight of the vector \(u_3.z\)  is zero. But as all the zero weight vectors in M lies in \(\overset{\circ }{W}\) , it follows that \(u_3 .z \in \overset{\circ }{W}\)  and hence \(z_1 = 0\)  which is a required contradiction.
This proves the subclaim. Now, as \(\psi (z_1) = \bar{z_1}\)  is a non-zero element of \(\overline{Z}\)  and \(\overline{Z}\)  is an irreducible \(\mathfrak {L}_{S,n}[K]\) -module, there exists an element \(u_1 \in U(\mathfrak {L}_{S,n}[K])\)  such that \(u_1 . \bar{z_1} = \bar{z}\) . But this implies that \(z - u_1.z_1 \in M \cap \overset{\circ }{W}\subseteq M^{\prime }\) . So, we have \(z + M^{\prime } = u_1.z_1 + M^{\prime }\) , which is a contradiction to the assumption that \(M_1/M^{\prime }\)  is a proper submodule of \(M/M^{\prime }\) . This completes the proof of claim 3.
By Theorem REF , we have \(M/M^{\prime } \cong L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\) ,
where \(Z = U({\mathfrak {L}_{S, n}} [K]).\bar{z}^{\prime }\) , where \(\bar{z}^{\prime }\)  is image of \(z\)  under the natural map from \(M\)  to \(M/M^{\prime }\) .
Let \({\bf {g}}_2\in K\)  such that \(V_{\mathbf {\lambda }+j_0 {\mathbf {\beta }}+{\bf {g}}_2}\ne 0\) . As \(L_{\mathfrak {L}_{S,n}}(K,{\mathbf {\beta }}, Z)\)  is non-uniformly bounded there exists \({\bf {g}}_1\in K\)  and \(p\in \mathbb {N}\)  such that
\(\mathrm {dim}{(L_{\mathfrak {L}_{S,n}}(K,{\mathbf {\beta }}, Z))}_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1} )> (\mathrm {dim}\,S+n)\mathrm {dim} V_{\mathbf {\lambda }+j_0\mathbf {\beta }+{\bf {g}}_2}\) 
Let \(0\ne f\in B\) . Let \( M^{\prime \prime }_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1}\)  be a vector space compliment of \(M^{\prime }_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1}\)  in \(M_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1}\) . Then we have
\(\mathrm {dim}M^{\prime \prime }_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1}>(\mathrm {dim}\,\,S+n)\mathrm {dim}V_{\mathbf {\lambda }+j_0\mathbf {\beta }+{\bf {g}}_2}\) .
By above inequality there exists \(0\ne w_{f,p}^{{\bf {g}}_1}\in M_{\mathbf {\lambda }+(j_0-p)\mathbf {\beta }+{\bf {g}}_1}\)  such that
\({((\mathfrak {L}_{S,n})_{p\mathbf {\beta }+{\bf {g}}^{\prime }}\otimes f)w_{f,p}^{{\bf {g}}_1}=0}\) , where \( {\bf {g}}^{\prime } = {\bf g_2 - g_1}\) .
Let us choose \(N \in \mathbb {N}\)  be large enough so that \(\mathbf {\lambda }+ (p + j) \mathbf {\beta }+ K \ne \bf 0\)  for all \(j \ge N\) . Then we have
\({(\mathfrak {L}_{S, n})_{(p+j)\mathbf {\beta }+{\bf {g}}}.w_{f,p}^{{\bf {g}}_1}=0}\)  \(\forall j\ge N\)  and \(\forall {\bf {g}}\in K\) .
As \(L_{\mathfrak {L}_{S,n}}(K,{\mathbf {\beta }}, Z)\)  is irreducible, there exists \(u_1,\dots ,u_\ell \)  and \(h_1,\dots ,h_\ell \)  such that
\(h_1\cdots h_qu_1\cdots u_\ell .w_{f,p}^{{\bf {g}}_1} = \bar{z}^{\prime },\)
where
\(u_i\in (\mathfrak {L}_{S, n})_{(j_i){\mathbf {\beta }}+{\bf {g}}_i^{\prime \prime }}, 1\le i \le \ell , j_i>0, \sum _{i = 1}^{\ell }j_i = p, \,\, \mathrm {and}\,\, {\bf {g}}_i^{\prime \prime }\in K\,\, \mathrm {and}\,\, h_k\in (\mathfrak {L}_{S, n})_{{\bf {g}}_k^{\prime }},\)  \( {\bf g}_k^{\prime }\in K,\,\, \forall \,\, 1\le k\le q.\) 
But as \(M_{\mathbf {\gamma }}^{\prime } = \lbrace 0\rbrace \)  for all \({\mathbf {\gamma }} \in \mathrm {Supp} (\overline{Z})\) , we have \(h_1\cdots h_q u_1\cdots u_\ell .w_{f,p}^{{\bf {g}}_1} = z.\) 
Now, using
\({((\mathfrak {L}_{S, n})_{p\mathbf {\beta }+{\bf {g}}^{\prime }}\otimes f)w_{f,p}^{{\bf {g}}_1}=0} ={(\mathfrak {L}_{S, n})_{(p+j)\mathbf {\beta }+{\bf {g}}}.w_{f,p}^{{\bf {g}}_1}},\)  for all \(j \ge N\) ,
we have
\({((\mathfrak {L}_{S, n})_{(2p+j)\mathbf {\beta }+{\bf {g}}}\otimes f)w_{f,p}^{{\bf {g}}_1}} = [{(\mathfrak {L}_{S, n})_{(p+j)\mathbf {\beta }+{\bf {g-g}}_1}, {((\mathfrak {L}_{S, n})_{p\mathbf {\beta }+{\bf {g}}_1}\otimes f)}].w_{f,p}^{{\bf {g}}_1}}=0 \, \forall {\bf g} \in K.\) 
But then for all \(j\ge N\) ,
\(((\mathfrak {L}_{S, n})_{(2p+j)\mathbf {\beta }+{\bf {g}}}\otimes f)z=((\mathfrak {L}_{S, n})_{(2p+j)\mathbf {\beta }+{\bf {g}}}\otimes f)h_1\cdots h_qu_1\cdots u_\ell .w_{f,p}^{{\bf {g}}_1}=0\)  \(\forall {\bf g}\in K \)
by an induction on \(q + \ell \)  and using \(((\mathfrak {L}_{S, n})_{(2p+j)\mathbf {\beta }+{\bf {g}}}\otimes f).w_{f,p}^{{\bf {g}}_1}=0\)  \(\forall {\bf g}\in K, j\ge N.\)
 Case 2 : \({\bf 0} \in \mathrm {Supp}({\overline{W}})\) . In this case we have two subcases:
Subcase (i): \(\lbrace \mathbf {\lambda }+j_0\mathbf {\beta }+K\rbrace  \cap \mathrm {Supp} (\overline{W}) \ne \lbrace  \bf 0\rbrace \) .
Let \( \overset{\circ }{\overline{W}}[ \mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]\)  be a maximal trivial \(\mathfrak {L}_{S,n}[K]\) -module of
\(\overline{W}[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]\) . It follows that
\( \overset{\circ }{\overline{W}}[ \mathbf {\lambda }+ j_0 \mathbf {\beta }+ K] = \overset{\circ }{{W}}[K] \cap W[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]/ \overset{\circ }{W} \cap W[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K],\) 
where \(\overset{\circ }{W}[K] : = \lbrace w \in W : \mathfrak {L}_{S,n}[K].w = 0\rbrace \) . Now, \(\overline{W}[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]/ \overset{\circ }{\overline{W}}[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]\)  contains a non-trivial irreducible \(\mathfrak {L}_{S,n}[K]\) -submodule, which we denote by \(\overline{D}\) .
We have
\(W[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]/ \overset{\circ }{W}[K] \cap W [\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K] \cong \overline{W}[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]/ \overset{\circ }{\overline{W}}[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K] \) 
as \(\mathfrak {L}_{S,n}[K]\) -modules.
Consider a natural surjective \(\mathfrak {L}_{S,n}[K]\) -module map
\(\Phi : W[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K] \rightarrow W[\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]/\overset{\circ }{W}[K] \cap W [\mathbf {\lambda }+ j_0 \mathbf {\beta }+ K]. \) 
Let \(0 \ne \bar{s} \in \overline{D}\)  be a non-zero vector of non-zero weight. Let \(s\)  be any preimage of
\(\bar{s}\)  under \(\Phi \) . Let \(T = U(\mathfrak {L}_{S, n}).s\)  and \(T^{\prime }\)  be its maximal \(\mathfrak {L}_{S,n}\) -module with the property that \(T^{\prime }_{\mathbf {\eta }} = 0\)  for all \(\mathbf {\eta }\in \mathrm {Supp}(\overline{D}) \setminus \lbrace 0\rbrace \) . It is easy to see that \(T \cap \overset{\circ }{W}[K] \subseteq T^{\prime }\) . Let us define an \(\mathfrak {L}_{S,n}[K]\) -module \(T\lbrace K\rbrace : = U(\mathfrak {L}_{S,n}[K]).s\)  and an \(\mathfrak {L}_{S,n}[K]\) -module surjective map \(\psi : T\lbrace K\rbrace  \mapsto \overline{D}\)  by \(\psi (u.s) = u.\bar{s}\) , where \(u \in U(\mathfrak {L}_{S,n}[K]).\)  Note that \(\psi \)  is a restriction of the map \(\Phi \)  on \(T\lbrace K\rbrace \)  and we have
\(T \lbrace K\rbrace / T\lbrace K\rbrace \cap \overset{\circ }{W}[K] \cong \overline{D}.\)
Claim 4: \(T/T^{\prime }\)  is an irreducible \(\mathfrak {L}_{S,n}\) -module.
Let us assume that the claim is false. So there exist a proper \(\mathfrak {L}_{S,n}\) -submodule \(T_1\)  containing \(T^{\prime }\)  such that \((T_1)_{\mathbf {\gamma }} \ne 0\)  for some \(\mathbf {\gamma }\in \mathrm {Supp}(\overline{D}) \setminus \lbrace \bf 0\rbrace \) . Let \(0 \ne s_1 \in (T_1)_{\mathbf {\gamma }}.\)  Then there
exists \(u \in U(\mathfrak {L}_{S,n})\)  such that \(u.s = s_1\) . We see here that \(u\)  must lie in \(U(\mathfrak {L}_{S,n}[K])\) ; otherwise \(u = u_1 u_2 u_3\)  where \(u_1 \in U_0(\mathfrak {L}_{S,n}^-)\) , \(u_2 \in U(\mathfrak {L}_{S,n}[K])\)  and \(u_3 \in U_0(\mathfrak {L}_{S,n}^+)\) . But we notice that
\(u_3. s = 0\)  unless weight of \(u_3. s\)  is zero. But this possibility is ruled out by the choice of \(j_0\)  in the equation REF . Hence we have \(u \in U(\mathfrak {L}_{S,n}[K])\) . Now, consider the image \(\psi (s_1) = \bar{s_1}\) , which is non-zero as \(s_1 \notin T^{\prime }\) . But then irreducibility of
\(\overline{D}\)  implies that there exists an \(u_1 \in U(\mathfrak {L}_{S,n}[K])\)  such that \(u_1.(\bar{s_1}) = \bar{s}\) . But this forces that \(s + T^{\prime } = u_1.s_1 + T^{\prime }\) , which is a contradiction to the assumption that \(T_1/T^{\prime }\)  is a proper submodule of \(T/T^{\prime }\)  and we are done with the proof of claim 4.
Now, we have \(T/T^{\prime } \cong L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, D)\) , where \(D = U(\mathfrak {L}_{S,n}[K]) \bar{s}^{\prime }\) , where \( \bar{s}^{\prime }\)  is the image of \(s\)  in \(T/T^{\prime }\) . Now, rest of the proof follows exactly along the similar lines as that of proof of case 1.
Subcase (ii): \(\lbrace \mathbf {\lambda }+j_0\mathbf {\beta }+K\rbrace  \bigcap \mathrm {Supp} (\overline{W}) = \lbrace  \bf 0\rbrace \) . In this subcase notice that \(\overline{W}[\mathbf {\lambda }+j_0 \mathbf {\beta }+K] = \overset{\circ }{W}[K]/ \overset{\circ }{W} = \overline{W}_{\bf 0}\)  ,where
\(\overset{\circ }{W}[K] : = \lbrace w \in W : \mathfrak {L}_{S,n}[K].w = 0\rbrace \)  Now, let \(0 \ne \bar{w} = w + \overset{\circ }{W}\) . Let \(j_1\)  be the largest integer with the property that \(j_1 < j_0\)  and \(\lbrace \mathbf {\lambda }+j_1\mathbf {\beta }+K\rbrace  \bigcap \mathrm {Supp} (\overline{W}) \ne \phi \) .
But using the property of the Lie algebra \(\mathfrak {L}_{S,n}\)  that \([(\mathfrak {L}_{S,n})_{\bf m}, (\mathfrak {L}_{S,n})_{\bf s}] = (\mathfrak {L}_{S,n})_{\bf m + s}\)  for \({\bf m, s} \in \mathbb {Z}^n\)  and \({\bf m} \ne {\bf s}\) , it follows that \(j_1 = j_0 -1\) . For if \(\lbrace  \mathbf {\lambda }+ (j_0 - 1) \mathbf {\beta }+ K\rbrace  \bigcap \mathrm {Supp} (\overline{W}) = \phi \) ,
then it would imply \(\mathfrak {L}_{S,n}[- \mathbf {\beta }+ K].w = 0\) . But then using the before mentioned bracket property of \(\mathfrak {L}_{S,n}\) , it will follow that \(\mathfrak {L}_{S,n}^- .w = 0\) . As we already have \(\mathfrak {L}_{S,n}^+ .w = 0 = \mathfrak {L}_{S,n}[K].w\) , it follows that \(\mathfrak {L}_{S,n}.w = 0\) , which is a contradiction to the assumption
that \(w \notin \overset{\circ }{W}\) . Let \(x \in (\mathfrak {L}_{S,n})_{-\mathbf {\beta }+ k}\)  such that \(x.w \ne 0\) , for some \(k \in K\) . Now using the facts that \(\mathfrak {L}_{S,n}^+ .w = 0 = \mathfrak {L}_{S,n}[K].w\) , a quick calculation shows that \(\mathfrak {L}_{S,n}^+ . (x.w) = 0\) . Let \(z_1 := x.w\)  and consider the \(\mathfrak {L}_{S,n}[K]\) -module
\(Z_1 = U(\mathfrak {L}_{S,n}[K]).z_1\) . An easy check shows that \(\mathfrak {L}^+_{S,n}. y = 0\)  for all \(y \in Z_1\) . Observe that \(Z_1\)  must be a uniformly bounded \(\mathfrak {L}_{S,n}[K]\) -module. Otherwise, considering it as \(\mathrm {Vir}(\mathbf {\gamma }) \cap \mathfrak {L}_{S,n}[K]\) -module and taking its irreducible subquotient \(\overline{Z_1}\) , which has finite dimensional but non-uniformly bounded weight spaces, we get a unique irreducible quotient of \(\mathrm {Vir}(\mathbf {\gamma })\) -module
\(\mathrm {Ind}_{U((\mathfrak {L}_{S,n}[K] \cap \mathrm {Vir}({\mathbf {\gamma }}))+({\mathfrak {L}_{S,n}}^+ \bigcap \mathrm {Vir}(\mathbf {\mathbf {\gamma }})))}^{U(\mathrm {Vir}(\mathbf {\mathbf {\gamma }}))}\bar{Z_1},\) 
which does not have finite dimensional weight spaces. This will eventually imply that \(W\)  does not have finite dimensional weight spaces. Now, as \(Z_1\)  is uniformly bounded and \({\bf 0} \notin \mathrm {Supp}(Z_1)\) , it contains a non-trivial irreducible \(U(\mathfrak {L}_{S,n}[K])\) -submodule
say \(Z\) . Then \(M : = U(\mathfrak {L}_{S,n}).Z\)  is a highest weight \(\mathfrak {L}_{S,n}\) -module contained in \(W\) . Let \(M/M^{\prime }\)  be its unique non-trivial irreducible quotient. Now, we can proof that any \(0 \ne z \in Z\)  is an \(\mathfrak {L}_{S,n}(B)\)  highest weight vector by using the same argument as in case 1.
Let \(L_{\mathfrak {L}_{S,n}(B)}(K, {\mathbf {\beta }}, Z)\)  be an irreducible highest weight \(\mathfrak {L}_{S,n}(B)\) -module with finite dimensional weight spaces. The highest weight space \(Z\)  is an irreducible \((\mathfrak {L}_{S,n}(B))_K\) -module with uniformly bounded weight spaces. We aim to show that
\(L_{\mathfrak {L}_{S,n}(B)}(K, {\mathbf {\beta }}, Z)\)  is a single point evaluation \(\mathfrak {L}_{S,n}(B)\) -module. We start with the following:
Proposition 4.8  The \((\mathfrak {L}_{S,n}(B))_K\) -module \(Z\)  is a single point evaluation module.

In the view of Theorem REF , it is enough to prove that \((\mathfrak {L}_{S,n}(B))_K\)  is isomorphic to \((\mathfrak {L}_{S^{\prime },n-1}(B))\)  as a Lie algebra, where \(S^{\prime }\)  is some finite dimensional abelian Lie algebra. It is enough to prove that
\((\mathfrak {L}_{S,n}(B))_K\)  is isomorphic to the Lie algebra \(\mathcal {G}: = \mathrm {span}\lbrace D({\bf u , r}) \otimes b, s \otimes t^{{\bf s}} \otimes b^{\prime } \mid {\bf r, s} \in \mathbb {Z}^n, r_n = 0 = s_n, s \in S, b,b^{\prime } \in B\rbrace \)  as \(\mathcal {G}\)  is isomorphic
to \(\mathfrak {L}_{S^{\prime }, n-1}(B)\) , with \(S^{\prime } := S \oplus \)  by the assignments
\(D({\bf {u, r}}) \otimes b \mapsto D({\bf {u, r}}) \otimes b\)  for \({\bf u} \in n\)  and \({\bf r} \in \mathbb {Z}^n\)  with \(u_n = 0 = r_n\) , \(D({\bf {e_n, r}}) \otimes b \mapsto {\bf e_n} \otimes t^{{\bf r}} \otimes b\) , and \(s \otimes t^{{\bf s}} \otimes b^{\prime } \mapsto s \otimes t^{\bf s} \otimes b^{\prime }.\) 
Let \({\mathbf {\alpha _1}, \ldots , {\mathbf {\alpha }_{n-1}}}\)  be elements of \(\mathbb {Z}^n\)  such that \(K = \sum _{i = 1}^{n-1}{\mathbb {Z}{\mathbf {\alpha }_i}}\) , and let \({\mathbf {\alpha }_n} \in \mathbb {Z}^n\)  such that \(\mathbb {Z}^n = K \oplus \mathbb {Z}{\mathbf {\alpha }_n}\) . Let \(\lbrace \mathbf {\gamma _1}, \ldots ,\mathbf {\gamma _n}\rbrace \)  be the basis of
\(\mathbb {R}^n\)  which is dual to \(\lbrace {\mathbf {\alpha _1}, \ldots , {\mathbf {\alpha }_{n}}}\rbrace \) . Then the map \(\Phi : \mathcal {G} \rightarrow (\mathfrak {L}_{S,n}(B))_K\)  by
\(\Phi (t^{\bf {r}}d_i \otimes b) = D({\mathbf {\gamma }_i}, \sum _{i = 1}{r_i {\mathbf {\alpha }_i}})\)  and \(\Phi (s \otimes t^{{\bf {r}}} \otimes b) = s\otimes t^{\sum _{i = 1}{r_i {\mathbf {\alpha }_i}}} \otimes b\)  for \(1 \le i \le n\)  and \(r_n = 0\) . An easy checking shows that the map \(\Phi \)  is a Lie algebra isomorphism
and we are done.
Theorem 4.9 
\(L_{\mathfrak {L}_{S,n}(B)}(K, {\mathbf {\beta }}, Z)\)  is a single point evaluation \(\mathfrak {L}_{S,n}(B)\) -module.

By Proposition REF , \(Z\)  is an evaluation \((\mathfrak {L}_{S,n}(B))_K\) -module. So there is an algebra homomorphism \(\psi : B \rightarrow  such that \)\((D({\bf u,r})\otimes b).z=\psi (b)D({\bf u, r}).z, (s\otimes t^{\bf m}\otimes b^{\prime }). z= \psi (b^{\prime })(s\otimes t^{\bf m}).z,\)\(where \)u n, r,m K\( and \) b, b' B, s S\( and \) z Z\(. Also as \) LS, n (B)+\( acts trivially on \) Z\(, we have\)\((D({\bf u}, p_1 {\bf \beta } + {\bf r})\otimes b).z = 0 = (s\otimes t^{l_1 {\mathbf {\beta }} + \bf m}\otimes b^{\prime }).z,\)\(for all \) b, b' B\( and \) p1\( and \) l1\( are any positive integers. It implies that \)\((D({\bf u}, p_1 {\bf \beta } + {\bf r})\otimes (b - \psi (b))z = 0 = (s\otimes t^{l_1 {\mathbf {\beta }} + \bf m})\otimes (b^{\prime }- \psi (b^{\prime }))z,\)\(where \)u n, r,m K\( and \) b, b' B, s S, p1, l1 N\( .\)
Claim : For \({\mathbf {\alpha }}\in \mathbb {Z}^n, {\mathbf {\alpha }} = p {\mathbf {\beta }} + {\bf r} \)  with \( p \le -1\)  and \({\bf r} \in K\) ,
\((D({\bf u}, {\mathbf {\alpha }})\otimes (b-\psi (b))).z =0= ((S\otimes t^{\mathbf {\alpha }})\otimes (b-\psi (b))).z\)  for all \(b\in B\) .
We will prove the claim by proving that \((D({\bf u}, {\mathbf {\alpha }})\otimes (b-\psi (b))).z\)  is a highest weight vector of \(V\)  which does not lie in highest weight space, hence must be zero. We will use induction on \(p\) .
Let \(p=-1.\)  Let \({\bf v}\in \mathbb {C}^n, {\mathbf {\mathbf {\gamma }}} \in \mathbb {Z}^n, {{\mathbf {\mathbf {\gamma }}}} = l {\mathbf {\beta }} + {\bf s}\)  where \( l \in \mathbb {N}, {\bf s} \in K \)  and \(b^{\prime }\in B \)  . We have
\((D({\bf v, {\mathbf {\mathbf {\gamma }}}})\otimes b^{\prime })(D({\bf u}, { \mathbf {\alpha }})\otimes (b -\psi (b))).z = (D({\bf w}, {\mathbf {\alpha }} + {\mathbf {\mathbf {\gamma }}})\otimes (b^{\prime }b-b^{\prime } \psi (b))).z,\) 
where \({\bf w}=({\bf v}, { \mathbf {\alpha }} ){\bf u}-({\bf u}, { \mathbf {\mathbf {\gamma }}}){\bf v}.\)
Now, if \( p + l \ge 1\)  then \((D({\bf w}, {\mathbf {\alpha }} + {\mathbf {\mathbf {\gamma }}})\otimes (b^{\prime }b-b^{\prime }\psi (b))).z = 0\) . In the case \(p + l =0 \) . We have
\((D({\bf w}, {\mathbf {\alpha }} + {\mathbf {\mathbf {\gamma }}})\otimes (b^{\prime }b-b^{\prime }\psi (b))).z = \psi (b^{\prime }b-b^{\prime }\psi (b))(D({\bf w}, {\mathbf {\alpha }} + {\mathbf {\mathbf {\gamma }}}).z =0.\) 
Similarly, we have
\((S\otimes t^{\mathbf {\mathbf {\gamma }}}\otimes b^{\prime }) (D({\bf u}, { \mathbf {\alpha }})\otimes (b -\psi (b))).z = 0.\) 
Now, it is easy to prove using induction that the above identity holds for all \(p \le -1\) .
It follows that \((D({\bf u, {\mathbf {\alpha }}})\otimes (b-\psi (b))).z\)  is a highest weight vector, hence must be zero.
Similarly we also have \(((S\otimes t^{\mathbf {\alpha }})\otimes (b-\psi (b))).z =0 \)  for all \({\mathbf {\alpha }} \in \mathbb {Z}^n\)  with \( p \le -1\) , \(b\in B\) .
Consider the set
\(T : = \lbrace v\in V~|~ (D({\bf u,r})\otimes (b-\psi (b))).v =0=((S\otimes t^{\bf m})\otimes (b-\psi (b))).v ~~ \forall b\in B, {\bf r,m}\in \mathbb {Z}^n \rbrace .\)
By above it follows that \(T \ne 0\)  as \(Z \subseteq T\) . It is an easy checking that \(T\)  is \(\mathfrak {L}_{S,n}(B)\) -submodule of \(V\) , hence must be \(V\) .
In the light of the above theorem we concentrate on \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\) , an irreducible highest weight \(\mathfrak {L}_{S,n}\) -module with finite dimensional weight spaces. We have the following.
Theorem 4.10 For an \(\mathfrak {L}_{S,n}\) -module \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\) , there exists a co-dimension one subspace \(\overset{\circ }{S}\)  of \(S\)  such that \(\overset{\circ }{S} \otimes A_n\)  acts trivially on \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\) . In particular,

If \(S \otimes t^{\bf 0}\)  acts nontrivially, then \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\)  is an irreducible highest weight module for \(\mathfrak {L}\) .

If \(S \otimes t^{\bf 0}\)  acts trivially, then \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\)  is an irreducible highest weight module for \(W_n\) .


By application of Proposition  and Theorem REF , there exists a co-dimension 1 subspace \(\overset{\circ }{S}\)  such that \( \overset{\circ }{S} \otimes t^{\bf m}\)  acts trivially on \(Z\)  for all \({\bf m} \in K\) . We also have
\(\overset{\circ }{S} \otimes t^{\mathbf {\alpha }}.Z=0\) 
for all \({\mathbf {\alpha }}\in \mathbb {Z}^n\)  such that \({\mathbf {\alpha }} = p {\mathbf {\beta }} + {\mathbf {s}}, p \in \mathbb {N}, \mathbf {s} \in K \) .
Following the same method as that of proof of Theorem REF  we get
\(\overset{\circ }{S} \otimes t^{\mathbf {\mathbf {\gamma }}}.Z=0\) 
for all \({\mathbf {\mathbf {\gamma }}}\in \mathbb {Z}^n, {\mathbf {\mathbf {\gamma }}} = \ell {\mathbf {\beta }} + {\bf s}, \ell \in \mathbb {Z}_{< 0}, {\bf s} \in K\) 
and the set
\(\lbrace v\in L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)~ | \,\,\,\, \overset{\circ }{S} \otimes t^{\bf r}.v=0 ~\forall {\bf r}\in \mathbb {Z}^n \rbrace \) 
is a non-zero submodule of \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\) , hence must be \(L_{\mathfrak {L}_{S,n}}(K, {\mathbf {\beta }}, Z)\)  itself. This completes proof of
\((1)\) . Proof of \((2)\)  follows by arguments similar to that of \((1)\) .

Final result: Non-uniformly bounded case
We sum up the results of this section in the following:
Theorem 4.11 
Let \(V\)  be an irreducible Harish-Chandra \(\mathfrak {L}_{S,n}\) -module with non-uniformly bounded weight spaces, then \(V\)  can be considered as an irreducible \(\mathfrak {L}(B)\)  or \(W_n (B)\) -module. Further, considered as an \(\mathfrak {L}(B)\)  or \(W_n (B)\) -module, V
is a single point evaluation module and upto a change of variables \(V \cong L_{\mathfrak {L}}(K, {\mathbf {\beta }}, Z)\)  or \( L_{W_n}(K, {\mathbf {\beta }}, Z)\) .


Applications and remarks
We start with following:
Definition 4.12 Let \(V\)  be any \(\mathfrak {L} (B)\) -module. An element \(v\in V\)  is called \(\mathfrak {L}\) -trivial if \(\mathfrak {L}.v = 0\) .

Proposition 4.13 
Let \(V\)  be a non-trivial irreducible \(\mathfrak {L}(B)\) -module with finite dimensional weight spaces. Then \(V\)  does not contain any non-zero \(\mathfrak {L}\) -trivial vector.

Let us begin on the contrary that there exists \(0 \ne w \in V\)  such that \(\mathfrak {L}.w = 0\) . Then there must exist \({\bf u}_1 \in n, {\bf r}_1 \in \mathbb {Z}^n\)  and \(b \in B\)  such that \(D({\bf u_1, \bf r}_1)\otimes b. w \ne 0\) .
Let \(W := \mathrm {span}\lbrace D({\bf u, \bf r}) \otimes b.w : \forall \,\, {\bf u} \in n, {\bf r} \in \mathbb {Z}^n\rbrace \) . It is trivial checking that \(W\)  acquires a \(W_n\) -module structure and the map
\(\Psi : \mathrm {ad}\,W_n \rightarrow W\)  defined by \(\Psi (D({\bf u, \bf r})) = D({\bf u, \bf r})\otimes b.w\)  is a \(W_n\) -module map. Now, as \(\Psi \)  is a non-zero map, the irreducibility of \(\mathrm {ad}\, W_n\)  implies
that \(\Psi \)  is an isomorphism. But then we have \(\mathrm {Supp} (V) = {\mathbf {\gamma }} + \mathbb {Z}^n,\)  where \({\mathbf {\gamma }}\)  is the weight of \(w\) . An immediate application of Theorem REF  implies that \(V\)  must be a uniformly bounded
\(\mathfrak {L}(B)\) -module. But then appealing to Theorem REF  we get that \(D({\bf u, \bf r}) \otimes b^{\prime }.w = \eta (b^{\prime })D({\bf u, \bf r}).w = 0\)  for all \({\bf u} \in n, {\bf r} \in \mathbb {Z}^n, b^{\prime } \in B\)  and \(\eta \)  is the algebra homomorphism from \(B\)  to \(. In particular we have \) D(u, r) b.w=0\( for all\)u n, r Zn\(, which is a required contradiction.\) 
We have the following result which generalises the result by Savage [15]}:
Corollary 4.14 Every irreducible Harish-chandra module for \(W_n (B)\)  \((n \ge 2)\)  is either uniformly bounded or a highest weight module. Further, every irreducible Harish-Chandra \(W_n (B)\) -module is a single point evaluation module.

Taking the abelian Lie algebra \(S\)  to be zero, the result is now immediate from Theorem REF  and Theorem REF .
Remark 4.15 .
We would like to mention the significant difference between classification result of irreducible non-uniformly Harish-Chandra module for \(n =1\)  and \(n \ge 2\) . Unlike the case \(n = 1\)  where irreducible non-uniformly bounded
Harish-Chandra modules are tenor product of finitely many irreducible single point generalised evaluation highest weight modules, for \(n \ge 2\) , every irreducible Harish-Chandra module (uniformly bounded or non-uniformly bounded) is a single point evaluation module.

Acknowledgments: Authors would like to thank Rencai Lu for some helpful discussions. First author would also like to thank Sudipta Mukherjee for some helpful discussions.
